WO R K I N G PA P E R S E R I E S
N O 6 7 6 / S E P T E M B E R 2 0 0 6
THE DISTRIBUTION OF
CONTRACT DURATIONS
ACROSS FIRMS
A UNIFIED FRAMEWORK
FOR UNDERSTANDING
AND COMPARING
DYNAMIC WAGE AND
PRICE SETTING MODELS
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W O R K I N G PA P E R S E R I E S
N O 6 7 6 / S E P T E M B E R 2 0 0 6
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THE DISTRIBUTION OF
CONTRACT DURATIONS
ACROSS FIRMS
A UNIFIED FRAMEWORK
FOR UNDERSTANDING
AND COMPARING
DYNAMIC WAGE AND
PRICE SETTING MODELS
1by Huw Dixon
2© European Central Bank, 2006 Address
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C O N T E N T S
Abstract 4
Non-technical summary 5
1 Introduction 8
2 Steady state distributions of durations
across firms 11
2.1 The distribution of completed
durations across firms 13
2.2 Evidence from micro data: prices are
stickier than we thought! 16
2.3 Examples 17
3 Pricing models with steady state
distributions of durations across firms 20
3.1 Modelling a multiple Calvo economy as a generalised Taylor and generalised
Calvo economy 22
4 The typology of contracts and aggregation 23
5 The forward lookingness of pricing rules 25
6 Price data: an application to the Bils-Klenow
data set 30
6.1 Pricing models compared 31
6.2 Alternative interpretations of the
BK data set 33
7 Conclusion 36
8 Bibliography 38
9 Appendix 41
Figures 49
Abstract
This paper shows how any steady state distribution of ages and related hazard rates can be represented as a distribution across …rms
of completed contract lengths. The distribution is consistent with
a Generalised Taylor Economy or a Generalised Calvo model with duration dependent reset probabilities. Equivalent distributions have di¤erent degrees of forward lookingness and imply di¤erent behaviour in response to monetary shocks. We also interpret data on the pro-portions of …rms changing price in a period, and the resultant range of average contract lengths.
JEL: E50.
Keywords: Contract length, steady state, hazard rate, Calvo, Tay-lor.
Non-Technical Summary.
The concept of nominal rigidity captures the frequency with which wage
or price-setters (hereafter “firms”) review and reset wages and prices. In
a perfect market, wages and prices will be perfectly flexible and vary so
as to be at their optimal level reflecting the current demand and cost
conditions facing the firm. The current generation of models used in
analyzing the effects of monetary policy (the new neoclassical synthesis
models) adopt a model of inertia based on a variety of dynamic pricing
models. The most commonly used pricing models are the Taylor model of
staggered contracts of a fixed and known duration and the Calvo model
where the contract length is random (both assume that the wage or price
is fixed over the contract length).
The purpose of this paper is to provide a unified framework for
modelling the distribution of contract durations which will be
comparable across different types of dynamic pricing models and which
will give a guide as to how to interpret empirical data.
If we look at an economy, we will observe some wages and prices
changing in each period and a resultant distribution of durations. In
steady state this distribution is the same in each period. There are four
different ways of looking at the same steady state distribution of contract
durations. First, we can take a census at a point in time and look at how
long the wages and prices at that moment have been in force: that is, we
measure the age distribution of contracts. Second, we can look at the
hazard rates for various ages, the proportion of contracts which terminate
at each age. Third, we can look at the population of contracts and their
distribution. Lastly, we introduce the new concept of
the distribution of
contracts durations across firms
DAF
. This takes the same cross-section
at a moment in time as the age distribution, but looks at the distribution of
completed contract lengths associated with it, rather than the age. The
average age of contracts in cross section will be about half the completed
lifetime.
contracts we get 8/5 priods. However, if we average across firms we get
the much longer 3 periods. I believe that many existing studies use the
statistical methodology developed for the distribution of contracts not for
the
DAF
, and hence have not captured the correct measure of nominal
rigidity.
In this paper, we develop a series of identities by which from any
of the four ways of representing the steady state you can recover the
other three (
Proposition 1 and Corollary 1).
We then link the statistical models to theoretical pricing models.
Firstly, we generalise the standard Taylor model to allow for a
distribution of known contract lengths, the Generalised Taylor Economy
(
GTE
): this corresponds to the
DAF
. Secondly, the Generalised Calvo
model with duration dependent reset probabilities corresponds to the
Hazard rate representation. We are then in a position to compare
different pricing models for the same distribution of contract lengths. We
use the example of the Multiple Calvo model (one where there are many
sectors, each with a sector specific Calvo reset probability), showing how
the same distribution can be modelled as a GTE (where firms know the
contract length ex ante) or as a Generalised Calvo model (where the
contract lengths are uncertain). We find that for a given distribution of
contract lengths, the GTE is more myopic than the GC or MC which put
the same weight on the future.
This paper argues that the DAF is the relevant way of
understanding nominal rigidity: because it is firms that set prices or
wages, we need to measure nominal rigidity across firms.
A commonly used framework for measuring nominal rigidity is to
look at the distribution of contract durations. A simple example will show
the
DAF
differs from this. Suppose we have two firms: F1 sets a
different price each period, F4 once every four periods. Suppose we take
a four period sample. We will observe 5 contracts or price-spells: four
for firm F1 and one for F4
1. If we take the average contract length across
mean contract length across contracts, you get only 2.7 quarters: this is
far too short and means that there is over-sampling of short contracts. We
also ask what is the shortest mean contract length consistent with a given
proportion of firms changing prices: we find that this is the reciprocal of
the proportion (proposition 3) and that the maximum mean is proportional
to the longest contract length (Proposition 4). Lastly, we generate the
Impulse-response functions (
IR
) for a simple model and compare them
for the B-K distribution: we find the
IR
of the
MC
and the
GC
are similar;
that of the
GTE
reflects is more myopic pricing rule.
The majority of theoretical calibrations and existing empirical
work on nominal rigidity are using the wrong statistical framework for
measuring nominal rigidity. The identities put forward in this paper
provide a fuller understanding of how pricing models and data can be
linked together.
The methodology is applied to the Bils-Klenow (2004) data set
wuich gives the average proportion of firms resetting prices for 354
categories used in the US CPI for the period 1995-7. We find that the
mean contract length across firms is 4.4 quarters, which fits in well with
the notion that the average duration of a price is 4 quarters and is the
correct measure of nominal rigidity. If on the other hand you look at the
1
Introduction
Dynamic pricing and wage-setting models have become central to macroeco-nomic modelling in the new neoclassical synthesis approach: they are the way that nominal rigidity is introduced into the macroeconomic system1. The concept of nominal rigidity captures in some sense the frequency with which wage and/or price setters reset or review wages and prices. It has be-come apparent that di¤erent models of pricing have di¤erent implications for matters such as the persistence of output, prices and in‡ation to monetary shocks. The question arises as to how the implications of models are linked to the frequency with which wages or prices are reset and the resultant distri-bution of contract lengths (where contract length refers to a completed wage or price spell). In this paper I develop a uni…ed approach which can be used to understand and compare the distribution of contract durations implied by models of price and wage-setting, and also empirical data on pricing. The key point is that since we seek to understand nominal rigidity, we want to measure the degree of price stickiness across wage and price setters ("…rms" hereafter): we want to know the average frequency of price adjustment and mean duration of price or wage spells at the "…rm" level. Much of existing work on nominal rigidity has adopted a statistical framework that does not take into account that prices are set by …rms, but treats each price spell as an entity in itself: this gives rise to a mis-measurement of nominal rigidity and has resulted in failing to compare di¤erent pricing models consistently (see for example Dixon and Kara 2006a).
We start from the idea of modelling the class of all steady state distribu-tions of duradistribu-tions across a given population (in this case, the …rms or unions that set prices or wages): we call this the distribution of durations across …rms (DAF). In steady state there are three equivalent ways of interpreting the distribution of durations. First there is the cross-sectional distribution of ages: how long has the price or wage contract lasted until now? This is like the population census. Note that since there is one price set by each …rm, by taking a cross-section we are simultaneously …nding the distribution of ages across current price spells and across …rms. Second, we can look at the distribution in terms of survival probabilities: from the cross-section of ages, what is the probability of progressing from one age to the next one. A
1See Goodfriend and King (1997), Erceg (1997), Clarida, Gertler and Gali (1999), Ascari (2000) and Huang and Liu (2002) inter alia..
third perspective is to take a cross-section of contracts that start at a point in time, and …nd the distribution of completed contract lengths to which this gives rise. In e¤ect, this last distribution is not taken across all …rms, but only across those …rms which reset wage or price: it is essentially a co-hort distribution. Lastly, we can look at the cross-section of contracts in steady-state across …rms and ask what is the distribution of completed con-tract lengths (lifetimes) of this cross-section: in e¤ect we are associating each …rm with a (completed) contract length to get a distribution of completed contract lengths across …rms. The …rst three concepts (distribution of ages, hazard rates and distribution across new starters) are very well understood in statistics, being basic tools in demography, evolutionary biology and else-where. The last concept, the distribution of completed durations across the population of …rms is a more novel concept, but it is the key to under-standing dynamic wage and price setting models since it is this concept that determines nominal rigidity. The contribution of this paper is to provide a set of steady state indentities between these four di¤erent ways of looking at the underlying steady state distribution: we can start from any of the …rst three concepts to arrive at the fourth which corresponds to nominal rigidity. In order to understand the di¤erence between the DAF and the distri-bution across new contracts, we can consider the following simple example. Four …rms reset price every month. Another four …rms reset price once a year: one …rm in each quarter. Suppose we take the cohort of …rms that reset price in January. There are 5 …rms resetting prices: the average length of contracts from this cohort is 8=3months. If we look at the distribution of contract lengths across all …rms, the average duration is7 months. Clearly, if we want to obtain a sensible measure of nominal rigidity, what matters is the behaviour of all …rms which indicates that we should use theDAF; not the distribution across new contracts. However, in calibrating the Calvo model, existing studies tend to use the mean of the distribution across new contracts, not across all …rms: hence the notion that a reset probability of 0:25 corresponds to an average contract length of 4 quarters - the average across new contracts is indeed 4 quarters, but across all …rms is 7 quarters.
We link the statistical analysis of contracts across …rms to general models of price and wage setting. First, the Generalized Taylor Economy (GT E) introduced in Kara and Dixon (2005), which starts from the distribution of completed contract lengths (lifetimes). There are many sectors, each with sector speci…c contract lengths. The simple Taylor economy where all con-tract lengths are the same is a special case of theGT E. Secondly, adopting
the hazard rate perspective, we take the Calvo approach, where contract lengths are stochastic with a reset probability which may be constant (as in the classical Calvo model) or duration dependent (Wolman 1999). We show that the Calvo model with duration dependent reset probabilities (denoted as theGeneralized Calvo model GC) is coextensive with the set of all steady state distributions: each possible steady state age distribution has exactly one GC and one GT E which corresponds to it. Hence, using this frame-work, we are able to compare the di¤erent models of pricing for any given distribution of durations across …rms. This enables us to isolate the precise e¤ect of the pricing model as opposed to the di¤erence in the distribution of contract lengths. As Dixon and Kara (2006a) showed, existing compar-isons of simple Taylor and simple Calvo models of pricing have failed to even ensure that the mean contract lengths are the same, let alone the overall distribution of contract lengths across …rms (see for example Kiley 2002).
It is widely recognized that there is a variety of pricing or wage-setting behaviour in most economies. This raises the question of aggregation: if we seek to represent the economy with a particular model, is the model itself consistent with this heterogeneity? This paper shows that both the GT E
and GC are closed under aggregation: if we combine two economies repre-sented by a GT E, the resultant economy will also be a GT E. Likewise the GC. More importantly, we show that this is not the case for either the simple Taylor or Calvo models. If there is heterogeneity in the econ-omy, then it cannot consistently be represented as a simple Taylor or Calvo process (except possibly as a dubious approximation). However, another generalization of the Calvo idea, the Multiple Calvo economy M C is closed under aggregation. In the M C economy, there are many sectors, each with a sector speci…c reset probability.
Given that we have a particular distribution of durations, what di¤erence does the pricing model make? Following the analysis of Dixon and Kara (2005), the concept of Forward Lookingness (F L) is employed: how far on average do agents look forward (what is the weighted mean number of periods price setters look forward when the set their price?). We …nd that in the
GT E model, …rms on average are more myopic than in the GC model for a give distribution of durations. This leads to observable di¤erences in impulse response functions in response to monetary shocks.
We also apply this approach to the Bils-Klenow data set (Bils and Klenow 2004). From the sectoral data for the proportion of changes in prices per month we are able to construct the average length of contracts under the
hypothesis that there is a calvo process in each sector, and also …nd that the shortest possible mean duration is achieved by the assumption that there is the simplestGT E consistent with the observed proportion, which consists of one or two consecutive contract durations that yield the observed proportion of prices changing (Proposition 3). The longest possible mean is proportional to the longest possible contract length (Proposition 4). This paper provides not only a simple and transparent discrete time framework for understanding nominal price rigidity in dynamic macromodels, but also indicates how em-pirical evidence from price data can be applied in a consistent and relevant manner.
In section 2 we review the well known facts about the steady state dis-tribution of ages and hazard rates and durations across new contracts. We then introduce the new concept of the distribution of durations across …rms and show how all four concepts are related by simple identities which are spreadsheet friendly. In section 3, we link the concepts to di¤erent models of pricing. In section 4 the issue of aggregation is considered. In section 5 we analyze the di¤erent pricing models in terms of forward lookingness and compare the mean reset prices. In section 6, we implement these ideas using the Bils-Klenow data set.
2
Steady State Distributions of Durations across
Firms.
We will consider the steady-state demographics of contracts in terms of their durations. Thelifetime of a contract is how long it lasts from its start to its …nish, a completed duration. The age of a contract at time t is how long it has been in force since it started. The age is a duration which may or may not be completed. We will …rst review the well known representation of steady state durations by the related concepts of theage distribution and
hazard rates (see for example, Kiefer 1988).
There is a continuum agents (we will call them …rms here) f which set wages or prices represented by the unit interval f 2[0;1]: In steady state we can take a cross-section at timet and measure the age distribution2: s j is the proportion of …rms which have contracts age j; s
j F
j=1 where F is
the oldest age in steady state3. In steady state, the distribution of ages is monotonic: you cannot have more older people than younger, since to become old you must …rst be young. Hence the set of all possible steady state age distributions is given by the following subset of the (F 1) unit simplex
F 1 :
F 1
M =
s
2 F 1 : sj sj+1
An alternative way of looking at the steady state distribution of durations is in terms of the hazard rate. The hazard rate at a particular age is the proportion of contracts at ageiwhich do not last any longer (contracts which end at agei, people who die at agei). Hence the hazard rate can be de…ned in terms of the age distribution: given the distribution of ages in steady-state
s 2 F 1
M ;the corresponding vector of hazard rates4 !2[0;1)F 1 is given by: !i= s i si+1 s i ;i= 1:::(F 1) (1)
Corresponding to the idea of a hazard rate is that of the survival proba-bility, the probability at birth that the price survives for at least i periods, with 1 = 1and fori >1
i = i=11(1 ! )
and we de…ne the sum of survival probabilities and its reciprocal!: =PFi=1 i != 1
Clearly, we can invert (1): we have F 1equations. Hence:
Observation 1 given!2[0;1)F 1;there exists a unique corresponding age pro…le s2 F 1
M given by:
s
i =! i i= 1:::F.
3In some theoretical applications such as the Calvo model of pricing, there may be in…nite lifetimes. The analysis presented is consistent with that, although for all practical applications a …nite maximum is required.
4Since the maximum length isF;without loss of genersality we set !
F = 1. If!i= 1
for some i < F, then i is the maximum duration and subsequent hazard rates become irrelevant. This leads to trivial non-uniqueness. We therefore de…ne F as the shortest duration with a reset probability of1:Hence fori < F; !i2[0;1):
Given the ‡ow of new contracts, the proportion surviving to age i is i : ! = 1 ensures adding up. From observation 1 s1 = !. From the de…nition of hazard rates and Observation 1 we can move from an age distribution s2 FM 1 to the hazard pro…le and vice versa.5
2.1
The Distribution of Completed Durations across
Firms.
Given a steady-state age distribution s 2 FM 1, we can ask what is the corresponding distribution of completed durations or lifetimes across …rms
2 F 1. Note, we are asking for the distribution across …rms (DAF). There is a unit interval of …rms: each …rm sets one price. When we measure the population shares i, we are measuring across …rms, just as we do when we take the age distribution. We are seeking to answer the question "what is the distribution of completed across the population of …rms". The popu-lation of …rms does not vary over time6, and that whilst some …rms change price frequently and some infrequently, each individual …rm over time has an average contract length, and the average in the economy is the average over the stock of …rms. It is this that corresponds to the concept of price-stickiness7.
We now derive the main results of the paper, which provides a framework of steady state identities which links the three familiar concepts of age dis-tribution, hazard rate and distribution across contracts to the distribution of completed contract lengths across …rms, and hence to nominal rigidity of prices and wages. First, we consider the relationship between the Hazard pro…le and the distribution of contract durations across …rms.
Proposition 1. (a) Consider any Hazard pro…le!2[0;1)F 1: There exists a unique distribution of lifetimes across …rms corresponding to !; 2
F 1 where:
i =!:i:!i: i:i= 1:::F (2)
5 This relationship is one of the building blocks of Life Tables (Chiang 1984), which are put to a variety of uses by demographers, actuaries and biologists.
6This comes from the assumption of a steady state.
7 The DAF is similar to the concept of the distribution of contract lengths across workers employed by Taylor (1993, page 36) in his analysis of non-synchronised wage-setting with di¤erent contract lengths.
(b) Consider any distribution of contract lengths across …rms given by
2 F 1. There exists a unique hazard pro…le that will generate this distribution in steady state !2[0;1)F 1 where:
!i= i i F X j=i aj j ! 1
All proofs are in the appendix. Now, since there is a 1-1 relationship between the hazard pro…le and the distribution of contract lengths across …rms, and in addition we know that there is a 1-1 relation between the hazard pro…le and the age distribution, it follows that there must be a 1-1 relationship between the age distribution and the distribution of completed contract lengths8.
Corollary 1 (a) Consider a steady-state age distribution s 2 FM 1. There exists a unique distribution of lifetimes across …rms 2 F 1 which corresponds to s, where
i = i si s
i+1 i= 1::F 1 (3)
F = F sF
(b) Given a distribution of steady-state completed lifetimes across …rms,
2 F 1, there exists a unique s 2 FM 1 corresponding to s j = F X i=j i i j = 1:::F (4)
Lastly, we can also ask for a given distribution of contracts across …rms
2 F 1; or age distribution s 2 MF 1or hazard pro…le ! 2 [0;1)F 1, what is the corresponding distribution of durations taken across the total population ofcontracts d 2 F 1:These are:
d i = i i:! (5a) = s i si+1 ! (5b) = !i: i (5c)
8Note that Corollary 1 is essentially a generalisation of Taylor’s 4 quarter example (1993, p37). Taylor’s notation for the distribution of ages is i 1 rather than si here:
The distribution of durations across contracts is the same as the distribution across …rms resetting prices. The more frequent price setters (shorter con-tracts) have a higher representation relative to longer contracts. Note that the rhs denominator of (5a) is the product of the contract length and the proportion of …rms resetting price. For the values of i < ! 1 , the share of the duration i is greater across contracts than …rms: for larger i > ! 1
the share across contracts is less than the share across …rms. Using equa-tion(5a), we can move simply between the distributions across contracts and across …rms. From(5a);the mean length across all contractsd (which also equals the life expectancy at birth) is:
d= F
X
i=1
i: di =! 1 (6)
The mean contract length taken over all contracts in steady state is the reciprocal of the proportion of …rms resetting price. This is precisely the "frequency based" estimate of mean contract length that has been commonly used empirically (see for example Bils and Klenow 2004, Barhad and Eden 2004) and also for calibration purposes in models of price-stickiness (for ex-ample, Clarida et al 1999 p.1666). However, it should be clear that this is the mean of the wrong distribution: it is the mean across …rms resetting price, not all …rms. There is clear length biased sampling: the shorter contracts (more frequent price resetters) are oversampled, resulting in an underestimate of nominal rigidity.
In the study of unemployment, each spell of unemployment is treated as an observation and the identity of the person involved is irrelevant. Hence the focus in the unemployment literature on the duration of unemployment has been on the ‡ow of new spells of unemployment and how long they will last, rather than on the stock of unemployed9. In demography and evolutionary biology, each duration corresponds to a single individual, hence since people only live once, the distribution across people is exactly the same as the distribution across individual durations: hence the focus here is also on cohort studies, exploring the distribution of ages, hazard rates and lifetimes across people born at the same time. The key di¤erence in this paper arises because we are interested in the pricing behaviour of…rms: hence whilst the
9However, Akerlof and Main (1981) did suggest using the average duration across all the unemployed as an important indicator (see the ensuing debate Carlson and Horrigan 1983, Akerlof and Main 1983).
‡ow of new contracts is of interest, the average duration of contracts across the stock of …rms is what determines price stickiness.
2.2
Evidence from micro data: prices are stickier than
we thought!
There are now several studies using micro price data: in particular the In-‡ation Persistence Network (IPN) across the Eurozone has been particu-larly comprehensive10. These studies adopt a common methodology using monthly micro CPI data across several countries. Here we will consider Alvarez and Hernando (2004) for Spain (covering 1994-2003), Veronese et al (2005) for Italy (covering1996-2003), Baudry et al (2004) for France (cov-ering 1994-2003). All these studies have data on individual products sold at individual outlets. Their terminology of a "price spell" (analogous to unemployment spell) is the same as our lifetime of a contract. They also havetrajectories for prices: this is the sequence of price spells for a product at an individual outlet. We can think of each trajectory as analogous to the sequence of price contracts for an individual …rm. These papers all provide estimates of the average length of a price spell: both across the population of all price spells (corresponding to d in equation 6) and also across trajec-tories, where a mean duration is calculated for each trajectory and then the average is taken across trajectories (corresponding to T). There are many detailed empirical issues to do with weighting, censoring and the introduction of the Euro which we ignore here. However, we can …nd the direct estimates of average durations taken across …rms (all durations are in months) and contracts:
Italy11: d= 8; T = 13: France12: d= 5:28; T = 7:24:
10See Dhyne et al (2005) for a summary of the IPN’s …ndings. 11Veronese et al (2005), Table A2.
12Baudry et al (2004) page 16. Note, the estimate of dis only for unweigthed data. A trajectory in the French data is does not correspond to our concept of a complete trajectory over the whole period: the average length of trajectories is only 17 months. Our T is taken to be theirTW: This will be anunderestimate, becuase there are broken
rather than complete trajectories.
These studies give the empirical distributions of contract lengths (price spells), but not across …rms (trajectories). The distributions are very skewed: there are many short spells and a very long tale of long spells. The mean length of price-spells is 2-3 quarters. The empirical evidence shows that when you take the distribution across …rms rather than across contracts, you get much longer average durations of 4-5 quarters (except for France) which is much more in line with the conventional wisdom and survey evidence that average durations are around 1 year.
2.3
Examples.
In this section we provide six examples of how our steady state indetities work. In the …rst column we state the reset probabilities (hazard rates)
f!ig; in the second and third the corresponding distribution of agesf sig and lifetimes f ig over …rms, and in the fourth the distribution di over contracts (…rms resetting prices). In the bottom row we compute the pro-portion of new contracts !, the average age of contracts s and the average lifetime T across …rms in steady state, and d the average lifetime across contracts14. Example 1 !1 = 109 s1 = 3740 1 = 9 10 d 1 = 3637 !2 = 0 s2 = 401 2 = 0 d 2 = 0 !3 = 0 s3 = 401 3 = 0 d 3 = 0 !4 = 1 s4 = 401 4 = 1 10 d 4 = 371 != 3740 s= 2320 T = 1310 d= 4037
In this example, there are two lengths of contracts: 90% are 1 period and 10% 4 periods. Note that d < s < T: because of the proliferation of short contracts, the mean lifetime across contracts is even less than the average age across …rms (in all the other examples, d > s).
13Alverez and Hernando (2004). T is taken from Table 6.1 Panel C. There is no direct measure of dusing the CP I weights (Panel A.gives the unweighted mean). The value quoted is derived from the inverse of the reset frequency for each sector aggregated using theCP I weights (page 13).
14Note that the value of! is computed directly from the hazards !
i; likewise dfrom d
i. The fact that!= s
Example 2 !1 = 14 s1 = 178 1 = 2 17 d 1 = 14 !2 = 12 s2 = 6 17 2 = 6 17 d 2 = 3 8 !3 = 1 s3 = 3 17 3 = 9 17 d 3 = 3 8 != 8 17 s= 29 17 T = 41 17 d= 17 8
This example has a gently rising reset probability, with the shares of completed contracts across …rms increasing with length of contract, as do the shares across contracts.
Example 3 !1= 14 s1= 3271 1 = 8 71 d 1 = 14 !2= 12 s2= 2471 2 = 24 71 d 2 = 38 !3= 34 s3= 1271 3 = 27 71 d 3 = 2796 !4= 1 s4= 713 4 = 12 71 d 4 = 323 ! = 3271 s= 12871 T = 18571 d= 7132
This is similar to example 2, with a rising hazard over four periods. The shares across …rms and contracts both peak at period 3 with a small 4-period share.
Example 4: Simple Taylor 4.
!1 = 0 s1 = 14 1 = d 1 = 0 !2 = 0 s2 = 14 2 = d 2 = 0 !3 = 0 s3 = 14 3 = d 3 = 0 !4 = 1 s4 = 14 4 = d 4 = 1 != 14 s= 52 T = d= 4
A simple lesson can be derived from example 4. When all contracts have the same length, the distribution across contracts is the same as the distribution across …rms.
Example 5: Taylor’s US economy We can now consider an example start-ing from an empirical distribution of completed contract lengths we can derive the corresponding GC: Taylor’s US economy represents the es-timated distribution of completed contract lengths15 (in quarters) in the third column. We can represent this in terms of the distribution of
15In fact, in Taylor (1993), the ages are estimated but not reported. In Table 2.2 page 48 the second column we believe to be the distributionf ig although it is reported as
ad
ages and duration dependent hazard rates (both to 4 Decimal places), distribution over contracts and the resultant averages.
!1 = 0:2017 s1= 0:3470 1 = 0:07 d1 = 0:2017 !2 = 0:3430 s2= 0:2770 2 = 0:19 d2 = 0:2738 !3 = 0:4213 s3= 0:1820 3 = 0:23 d3 = 0:1825 !4 = 0:4986 s4= 0:1052 4 = 0:21 d4 = 0:1513 !5 = 0:5682 s5= 0:0528 5 = 0:15 d5 = 0:0865 !6 = 0:5849 s6= 0:0228 6 = 0:08 d6 = 0:0384 !7 = 0:6038 s7= 0:0095 7 = 0:04 d7 = 0:0165 !8 = 1 s8= 0:0037 8 = 0:03 d8 = 0:0108 != 0:3470 s= 2:365 T = 3:730 d= 2:8818
It is interesting to note that here, unlike examples 1-4, we can really see the di¤erence between the distribution across contracts and across …rms: in the distribution across contracts durations 1 and 2 are really boosted - we see a lot of shorter contracts. All the other durations are reduced, and in particular the longer contract lengths are much less common in the distribution across contracts and across …rms. The resultant mean duration is 77% of the mean across …rms.
Example 6: Simple Calvo The Calvo model most naturally relates to the hazard rate approach to viewing the steady state distribution of dura-tions. The simple Calvo model has a constant reset probability! (the hazard rate) in any period that the …rm will be able to review and if so desired reset its price. This reset probability is exogenous and does not depend on how long the current price has been in place. We can think about a sequence of uninterrupted periods without any review as the "contract length". The distribution of ages of contracts is
s =!(1 !)s 1: s= 1:::1
which has means=P1s=1 s:s=! 1. Applying Proposition 1(a) gives us the steady-state distribution of completed contract lengths i across …rms:
i =!2i(1 !)i 1: i= 1:::1 (7)
to predominate". The third column which is reported as f ig is monotonic so may be
ages. We have not been able to …nd an interpretation of Table 2.2 which is consistent with the steady state identities in this paper.
which has mean T = 2! 1 1 (see Dixon and Kara 2006). Note that for the simple Calvo model, the distribution of ages is the same as the distribution across contracts: substituting (7) into (5) yields s
i = di
i = 1:::1, so that the mean age of contracts across …rms equals the mean lifetime across new contracts and is the reciprocal of the reset probability. We illustrate the simple Calvo model with != 0:25.
!1= 0:25 s1 = 0:25 1= 0:0625 d1= 0:25 !2= 0:25 s2 = 0:1875 2= 0:09375 d2= 0:1875 !3= 0:25 s3 = 0:1406 3= 0:10546875 d3= 0:1406 !4= 0:25 s4 = 0:1052 4= 0:10546875 d4= 0:1052 !i= 0:25 si = 0:25 (0:75) i 1 i= (0:25)2i(0:75)i 1 di = si != 0:25 s= 4 T = 7 d= 4
3
Pricing Models with steady state
distribu-tions of duradistribu-tions across …rms.
Having derived a uni…ed framework for understanding the set of all possible steady state distributions of durations across …rms, we can now see how this can be used to understand commonly used models of pricing behaviour based on generalisations of Taylor and Calvo.
The Generalised Taylor Economy GT E Using the concept of the Gen-eralised Taylor economy GT E developed in Dixon and Kara (2005a), any steady-state distribution of completed durations across …rms 2 F 1can be represented by the GT E with the sector shares given by
2 F 1 : GT E( ). In each sector i there is an i period Taylor contract, with i cohorts of equal size (since we are considering only uniform GT Es): The sector share is given by i: Since the cohorts are of equal size and there as many cohorts as periods, there are i:i 1 contracts renewed each period in sector i. This is exactly as required in a steady-state. Hence the set of all possible GT Es is equivalent to the set of all possible steady-state distributions of durations. It is simple to verify that the age-distribution in a GT E is given by(4). If we want to know how many contracts are at agedj periods, we look at sectors with lifetimes at least as large as j, i = j:::F. In each sector
i, there is is a cohort of size i:i 1 which set its price j periods ago. We simply sum over all sectors i j to get (4). The simple Taylor economy (ST) is a special case where there is only one sector and one contract length.
The Generalised Calvo model (GC). The representation of the steady-state distribution by hazard rates suggests generalising the Calvo model to allow for the reset probability (hazard) to vary with the age of the contract (a duration dependent hazard rate), as in Wolman (1999),. This we will denote the Generalised Calvo Model GC. A GC is de-…ned by a sequence of reset probabilities: as in the previous section this can be represented by any ! 2 [0;1)F 1. From observation 1, given any possible GC there is a unique age pro…le s 2 FM 1 corre-sponding to it and a unique distribution of completed contract lengths from Proposition 1. Again, Proposition1, if we have a distribution of completed contract lengths, there is a unique GC which corresponds to it. Thus, the two approaches to modelling pricing: the GT E and theGC are comprehensive and coextensive, both being consistent with any steady-state distribution of durations16.
The Multiple Calvo Model (M C). In this approach, the economy is seen of as comprising several sectors, each with a simple Calvo process (i.e. a sector speci…c hazard/reset probability). Alvarez et al (2005) argue that an aggregate hazard rate declines over time and that this can be attributed to the heterogeneity of hazard rates. We can de…ne a mul-tiple Calvo process M C as M C(!; ) where!2(0;1]n gives a sector speci…c hazard rate17 !
k for each sectork= 1; :::nand 2 n 1 is the vector of shares k.
We can thus see that all of the Above models generate a steady state distribution of durations. We can ask which pricing models do not yield a steady state. The important thing to note about the above models is that
16Note that an alternative parameterization of the duration dependent hazard rate model is to specify not the hazard rate at each duration, but rather the probability of the completed contract length at birth (see for example Guerrieri 2004). The probability at birthfi of a contract lasting exactly iperiods is simply the probability it survives to
periodiand then resets ati:fi= i!i:
17The notation here should not be confused: the substrcriptskare sectoral: none of the sectoral calvo reset probabiltities are duration dependent in theM C model.
the distrbution of contract lengths does not depend in any way upon the state of the economy: it is essentially the result of an an exogenous process which terminates contracts. Thus even though the …rm or the economy are not in steady state, the distribtion of durations is invariant. State dependent or menu cost models do not have this property (for example Dotsey et al 1999). When the economy is in steady state they do posses a steady state distribution of durations. However, the state of the economy a¤ects the duration of contracts, which means that when the economy is out of steady state (due to an exogenous shock of some kind), durations of contracts may also deviate from the steady state.
3.1
Modelling a Multiple Calvo economy as a
Gener-alised Taylor and GenerGener-alised Calvo economy.
Having identi…ed three di¤erent pricing models, we can use the uni…ed frame-work to show how we can move betweeen them. The example we will use is the M C model: we will show how to model this as a GT E and a GC with the same steady state distribution of durations across …rms. So, let us take as the starting point the multiple-Calvo porcessM C(!; ). To model this as a GT E, we can take the distribution of duration within each sector: let
kibe the proportion of iperiod contracts in thek;from (7)we have: ki =!ki(1 !k)i 1
The proportion ofiperiod contracts across the whole economy, iis obtained by summing accross sectors using the weights k
i = n
X
k=1
k ki (8)
Hence we can represent M C(!; )by GT E( ) using(8).
From this distribution of completed durations, we can construct the cor-responding GC from Proposition 1(b). The ‡ow of new contracts is ! and
the distribution of lifetimes or the underlying sectoral reset probabilities: ! = 1 X i=1 i i = 1 X i=1 Pn k=1 k ki i = 1 X i=1 n X k=1 k!k(1 !k)i 1 !i = i i 1 X j=0 ai+j i+j ! 1 = Pn k=1 k!k(1 !k)i 1 P1 j=0 Pn k=1 k!k(1 !k) i+j 1
Proposition 2: The aggregate GC model corresponding to M C model has a declining hazard rate. In the limit as i! 1, the hazard rate in the
GC tends to the lowest hazard rate in the M C.
Clearly, the aggregate hazard in the GC corresponding to an M C is decreasing over time: !i> !i+1. The way to understand this is that sectors with higher !k tend to change contract sooner. So, for a given cohort, the relative share of sectors with higher!k tends to go down. At any durationi, the share of typekcontracts increases if the reset probability is below average or decreases if it is above average. The reset probability gradually declines and asymptotically reaches the lowest reset probability. In the long-run, the type with the lowest reset probability comes to dominate and theGC tends to this lowest vale.
4
The Typology of Contracts and
Aggrega-tion.
In terms of contract structure, we can say that the following relationships hold:
GC =GT E = SS. The set of all possible steady state distributions of durations is equivalent to the set of all possible GT Es and the set of all possible GCs.
C M C GC. The set of distributions generated by the Simple Calvo is a special case of the set generated by M C which is a special case of GC.
ST GT E =GC Simple Taylor is a special case of GT E, and hence also of GC.
ST\M C =?. Simple Taylor contracts are a special case of GC, but not of M C.
Figure 1: The typology of Contracts
This is depicted in Fig 1. The GC and the GT E are coextensive, being the set of all possible steady-state distributions (Proposition 1). The Simple calvoC (one reset probability) is a strict subset of the Multiple Calvo process
M C which is a strict subset of the GC. The simple Taylor ST and the
M C are disjoint. The ST is a strict subset of the GT E: The size of the distributions is re‡ected by the Figure: ST has elements corresponding to the set of integers and is represented by a few dots; Calvo is represented by the unit interval; M C by the unit interval squared.
We can now ask the question: if we aggregate over two contract structures, what is the type of contract structure that results? This is an important question: if we believe that the economy is heterogenous, we should not represent it with a contract type which is not closed under aggregation. We can think of this in terms of giving each contract structure a strictly positive proportion of the total set of contracts; for example50%: We can de…ne the
ST in terms of contract length, under the assumption that each cohort is of equal size.
ST(k) +ST(j) =GT E((0:5;0:5);(k; j))
Clearly, if we aggregate over Standard Taylor contracts with di¤erent contract lengths k > j, we no longer have a Standard Taylor contract but aGT E.
Similarly, if we aggregate over simpleCalvocontracts with di¤erent reset probabilities, we do not get aC contract but a multiple Calvo M C:
C(!1) +C(!2) =M C((0:5;0:5);(!1; !2))
By de…nition, If we aggregate overM Cs, we still have anM C. We can say that a type of contract structure is closed under the operation of aggrega-tion if we aggregate two di¤erent contracts of that type and the resultant contract structure is also of the same type. Clearly, neither the ST or C
are closed under aggregation. However, M C; GC, and GT E are all closed under aggregation:
Consider the case of GT E. Suppose we have two GT Eswith maximum contract lengths F1 and F2 respectively and w:l.o.g: F1 F2: The corre-sponding vector of sector shares is then j 2 F2 1, j = 1;2 where we set 1i = 0fori > F1. If we combine the twoGT Es, we get anotherGT E with the sector shares being the average of the other two.
GT E( 1) +GT E( 2) =GT E 1
2 1+
1
2 2
Hence GT Es are closed under aggregation. Similarly, since GCs can be represented as equivalent GT Es; the closure of GT Es implies the closure of GC under aggregation.M C are closed, since we can simply combine the di¤erent!ifrom eachM Cand reweight on a50-50basis. We have illustrated the proof using two distributions with a 50-50 combination. The idea obvi-ously generalises to a convex combination of any number ofM C,GT E and
GCs:
The importance of this observation is that if we really do believe that contract structures are heterogeneous, we should use contract types that are closed under aggregation. The simple Calvo and Taylor models are only applicable if there is one type of contract and no heterogeneity in the economy. If we believe the Calvo model, but that reset probabilities are heterogenous across price or wage setters, then theM C makes sense. If we believe that the Calvo model is not a good one, then the GC or GT E is appropriate.
5
The Forward Lookingness of pricing rules.
We have developed a general framework for understanding steady-state dis-tributions of durations across …rms and how they are related in terms of pricing models. In this section we consider how pricing models di¤er when we control for the distribution of durations (requiring the steady state dis-tributions to be the same) in terms of the "Forward Lookingness" (F L) of the pricing rules. A pricing rule uses data from the present and future in order to determine the optimal price. In a log-lineraised form, this gives the current price as a linear function of data from each date ahead. The forward lookingness of a pricing rule takes the weights (normalised to unity) in the lin-earised decision rule and is the resultant average over the dates ahead. This simple measure capures the extent to which the future in‡uences the pricing
decision, and is applicable across all pricing rules.18. This concept was intro-duced in Dixon and Kara (2005) to compare the simple Calvo model and the equivalentGT E (named the Calvo-GT E). The Calvo-GT E is a model with exactly the same distribution of completed contract lengths as in the Calvo model, as given by(7b). We now generalise this to the GC framework. In this paper we ignore discounting, since it applies to all pricing rules: however, whilst it would be simple to generalise the formulae to allow for discounting, the no-discount case allows us to understand the di¤erences more clearly.
Let the optimal price (or target variable in general) in period t+s be
Pt+s so that the future information in‡uencing the pricing decision at t is
Pt+s 1s=0. In aGC with ! 2 [0;1)F 1; the proportion of …rms resetting price at time t is ! and they all set the same reset price XtGC: Ignoring discounting, the reset wage and forward lookingness19 F L are:
XtGC =! FX1 j=0 jPt+j 1 = F 1 X j=0 s jPt+j 1 (9)
The weights in theGC are the distribution of ages sj. This means that the forward lookingness of theGC is simply the average age of the distribution:
F LGC =!
F 1
X
s=1
s s=s sj =s (10)
In the corresponding GT E there are F sectors, with contract lengths
i= 1:::F with corresponding sector shares given by (2): i= i:!i si
18Note that Forward Lookingness is not in general equal to the expected duration of the contract when the price is set (life expectation at birth). They are equal in the simple Calvo model because it has the special property that life expectancy at birth equals the average age. For example, in the simple Taylor model without discountingF L=s, but life expectation at birth equals the full length of the contract (average completed contract lengthT):
19Note that Forward Lookingness is not in general equal to the expected duration of the contract when the price is set (life expectation at birth). They are equal in the simple Calvo model because it has the special property that life expectancy at birth equals the average age. For example, in the simple Taylor model, F L=s, but life expectation at birth equals the full length of the contract (average completed contract lengthT):
In each sector i, a proportion i 1 reset their prices every period: hence a total of! reset their prices, since from Proposition 1:
!= F X i=1 i i (11)
The important thing to note about (11) is that the longer contract lengths are under-represented amongst resetters relative to the population, since they reset prices less often. This means that the forward lookingness of those making the pricing decision is much less than the average contract length.
The reset price in sectori will be
Xit= 1 i i 1 X j=0 Pt+j
The average reset price across those resetting price is thus
XtGT E = 1 ! F X i=1 i i Xit= 1 ! F X i=1 i i2 i 1 X j=0 Pt+j = 1 ! F X i=1 !i si i i 1 X j=0 Pt+j = F X j=1 bsPt+j 1
where the weights on future informationbs can be written
bj = F X i=j i !i2 = F X i=j !i si !i (12)
Hence the forward lookingness of the GT E is
F LGT E = F
X
j=1
jbj
It is illuminating to write the GT E weights in terms of the GC weights. In theGT E, each price setter knows the exact length of the contact: hence
when setting price it ignores what happens after the end of the contract. In contrast, in the GC the price-setter is uncertain of the contract length and must always consider the possibility of lasting until the longest duration
F. As we identi…ed in Dixon and Kara (2005), this results in the fact that comparing theGT E to theGC weights, weight is "passed back" from longer durations to shorter making theGT E more myopic:
bj = !j ! s j j j+ 1 !j ! s j+ 1 X i=j+1 !i ! s i i (13)
There are three components of bj in (13): the corresponding sj, the weight passed back to more recent dates i < j and thirdly the weight it receives from longer contracts which include . To translate from s
j to bj, you need to correct by a factor of!j=! for each durationj: in the simple Calvo model this is unity and the equation reverts to Dixon and Kara (2005). This means that the bjs put a greater weight on the immediate future and less on the more distant future than the correspondingGC. We can see this if we look at the cumulative weights: looking at the sum of weights up to q periods ahead, the sum of bjs is the sum of Cjs plus the weights passed back from the periods in the future to each of thebj wherej q.
q X j=1 bj = q X j=1 !j ! j+ q X k=0 1 X i=k !i ! i i+ 1
The intuition behind this result is the following. In the GC, all …rms are uncertain about how long the price they set will last: all …rms have to take into account the optimal price for all F periods. This results in the optimal weights for each future period being the corresponding age shares. In theGT E, however, …rms only worry about the prices that cover the known contract length. Thus …rms with perfectly ‡exible prices can ignore the future when they set prices. Firms withiperiod contracts just lookiperiods ahead. The only people to worry about the optimal price F 1 periods ahead are the …rms with F period contracts. However, they also have to worry about all the prices along the way. Thus, when we consider (13), the bit "passed back" re‡ects the fact that the i period contracts have to take into account the periods up toi: the bit received from longer contracts is the corresponding bit they pass back to periodi.
We can illustrate the di¤erences in forward lookingness using the some of the examples we considered before in section 3.1. Since theGC weights are
just the age-shares s
i and were given previously as fractions in section 2.2, we state them here as decimals to 4 decimal places. Thebj coe¢ cients are give as both exact fractions and decimals.
Example 1 s 1 = 0:925 b1 = 0:9521 = 457480 s 2 = 0:025 b2 = 0:0271 = 48013 s 3 = 0:025 b3 = 0:0146 = 4807 s 4 = 0:025 b4 = 0:0063 = 1601 F LGC = 1:15 F LGT E= 486 480 = 1:075 Example 2. s 1 = 0:4706 b1 = 169 = 0:5625 s 2 = 0:3530 b2 = 165 = 0:3125 s 3 = 0:1765 b3 = 162 = 0:125 F LGC = 1:706 F LGT E= 25 16 = 1:5625 Example 3 s 1 = 0:4507 b1= 12871 = 0:5547 s 2 = 0:3380 b2= 12839 = 0:3047 s 3 = 0:1690 b3= 12815 = 0:1172 s 4 = 0:0423 b4= 1283 = 0:0234 F LGC = 1:803 F LGT E = 1:609
Clearly, in all three examples the GT E corresponding to a GC puts a much greater weight on the current period and less on the subsequent periods, resulting in a less forward looking pricing decision. In example 1, the weight is still greater on the second period, but falls o¤ rapidly.
Lastly, we can consider the case of a M C process. In this case, the forward lookingness of the M C is simply the average of the Forward look-ingnesses of the individual Calvo processes weighted by sector shares k
F LM C = n X k=1 k!k = k X j=1 ksk =s
wheresk is the average age in steady state in thek sector, andsthe average age in the population. Since by construction the M C and the equivalent
in the population, the two di¤erent constructions have the same forward-lookingness, F LGC = F LM C. Furthermore, we can see that the average reset price at timet will be equal.
In the M C, there will be di¤erent reset prices, one for each !k. Hence the reset price of …rms with hazard!k is:
XktM C = 1 X j=1 s kjPt+j 1 where s kj 1
j=1 is the steady-state age-distribution for those with hazard!k. The average reset price is then
XtM C = n X k=1 iXktM C = n X k=1 1 X j=1 k skjPt+j 1 = 1 X j=1 s jPt+j 1 since Pnk=1 k s kj = sj:
Hence the average reset price in theM Cis the same as the equivalentGC
setting. This means that when modelling an economy with heterogeneous Calvo contracts as in theM Cmodel, it may well be the most parsimonious to use the GC framework. The degree of forward lookingness and the average reset price are the same. The only di¤erence is that in M C there are as many reset prices as hazard rates, whereas in theGC there is only one.reset price in any one period.
6
Price Data: an application to the Bils-Klenow
Data set.
In this section, we apply our theoretical framework to the Bils-Klenow Data set (Bils and Klenow 1994). This data is the micro-price data collected monthly for the US CPI over the period 1995-7. The BK data covers 350 categories of commodities comprising 68.9% of total consumer expenditure. They focus on the proportion of prices that change in a month in each cate-gory (sector). They then derive the distribution of durations across contracts on the assumption that there is a sector speci…c Calvo reset probability in
continuous time. As we have argued, we belive that the distribution across contracts is not the right way to quantify price-stickiness.
In this section I use the BK data to construct the distribution of contract lengths across …rms. Each sector has a sector-speci…c average proportion of …rms resetting their price per month over the period covered. I interpret this as a Calvo reset probability in discrete time20. We adopt the discrete time approach in order to be consistent with the pricing models which are in discrete time. The …rst approach we adopt is to model this as a Multiple Calvo processBK M C. The second is to model the resulting distribution across all sectors. Within each sector we have the Calvo distribution of contract lengths as derived in Dixon and Kara (2006a): using the sectoral weights we can then aggregate across all sectors. This gives us the following distribution of contract lengths depicted in Figure 2:
Fig 2: the BK Distribution of Contract lengths Across Firms
Note that the mean of 4.4 quarters is larger than is reported in BK. This is because we are looking at the mean duration across …rms rather than contracts and hence we are more likely to observe longer contracts. With the aggregate distribution of contract lengths we can model this as either aGT E
or aGC as well as anM C. We therefore have three di¤erent pricing models of the same distribution of contract lengths derived from theBK dataset.
6.1
Pricing Models Compared.
We will see how the di¤erent models of pricing di¤er in terms of their impulse-response. We adopt the model of price or wage setting developed in Dixon and Kara (2005a, 2006): the details are set out in the appendix. We con-sider two monetary policy shocks: in the …rst case there is a one o¤ perma-nent shock in the level of the money supply; in the second an autoregressive
20The use of continuous time leads to a lower expected expected duration at birth. If the proportion resetting price is!, the expected duration at birth is 1=In(1 !). This is less than the discrete time expectation 1=!, partly re‡ecting the fact that the price might change more than once in a given period. The di¤erence gets proportinatley larger as! gets larger. When!= 0:8the discrete time estimator is over twice the continuous time estimator. The analysis in this paper is in discrete time becuase that is how the pricing models are employed in the literature, and also it provides spreadsheet simplicity and trnasparency.
process. Expressed as log deviations form steady state, money follows the process
mt = mt 1+"t
"t = "t 1+ t
where t is a white noise error term. We consider the case of = 0 and the autoregressive case = 0:5. The other key parameter captures the sensitivity to the ‡exible price to output. The optimal ‡exible price at periodt in any sectorpt is given by
pt =pt+ yt
where(pt; yt)are aggregate price and output (all in log-deviation form). We allow for two values of =f0:01;0:2g: a high one and a low one as discussed in Dixon and Kara (2006b).
Fig 3: Responses to a one-o¤ monetary Shock.
In Figure 3, we depict the responses of output, the reset price, the general price level and in‡ation to a one-o¤ shock with = 0:2. Looking at all the graphs, it is striking that the three models of pricing have fairly similar impulse-responses: none of them are far apart. However, in all cases the
M Cand theGCare close together and theGT E is farther away, particularly towards the end. To understand this, we can look at theIRfor the average reset price and the general price level. In the GT E case, the reset price rises less on impact than the M C or GC. This re‡ects the greater myopia: those cohorts resetting prices look less far ahead on average, so that they do not raise prices as much as in the M C or GC case. At about 10 months however, the situation is reversed: theGT E reset price exceeds theM C and
GC case: whilst the latter are slowing down price increases in anticipation of the approaching steady state, theGT E maintains momentum for longer. This comparative myopia of the GT E explains why the output response starts o¤ above both theM C and GC, but ends up after 15 months below both.
F ig 4: Serial Correlation in Monetary growth = 0:5
In Figure 4 we consider the autoregressive monetary policy shock and concentrate on the IR for output and in‡ation for both the high and the low values of . We …nd that there is now a more radical di¤erence between
the GT E and the other two models. If we look at in‡ation we see that there is a hump shape: the peak impact on in‡ation appears after the initial monetary shock: with the high value of it happens at 3 months: with the low value at around20 months. Both the M C and the GC are not hump shaped. This re‡ects the …nding in Dixon and Kara 2006b that the Calvo model does not capture the characteristic "hump shaped" response indicated by empirical VARS. This feature appears tp be shared by its generalisations
M C and GC.
This simple example of the IR of major variables shows how di¤erent models of pricing can yield di¤erent patterns of behaviour even though the distribution of contract lengths are exactly the same. Partly this is due to di¤erent degrees of forward lookingness. The M C and the GC do di¤er slightly, but are quite close, which re‡ects the fact that they have the same forward lookingness. It suggests that since theGC is computationally much simpler (you only have to model one pricing decision for all …rms resetting price, rather than one for each sector), this model might be preferred to the
M C.
6.2
Alternative Interpretations of the
BK
Data set.
The previous analysis was based on assuming that the true distribution within each sector is generated by the sector speci…c discrete time Calvo distribution. However, this is just one hypothesis about the underlying dis-tribution of contract lengths generating the proportion of …rms resetting their price per month. Let us look at the class ofGT Es that are consistent with a particular reset proportion. Again, let us consider the set ofGT Es with max-imum contract lengthsF; 2 F 1 :we can de…ne the subset which yield a particular reset proportion!: de…ne the mapping A(!) : [0;1]! F 1
A(!) = ( 2 F 1 : F X i=1 i i =! )
Note that since A(!) is de…ned by a linear restriction on the sector shares . We assume that ! F 1 in what follows, so that A(!) is non-empty21. HenceA(!) F 2andA(!)is closed and bounded. Note that if! = 1, all
21This is a purely technical assumption. If we assumed a value! < F 1, then even if all
…rms must reset their price every period, so that 1 = 1andA(!)collapses to a singleton. To avoid trivialities, we will restrict ourselves to!2[F 1;1). The average length of contracts is T( ) = PFi=1i: i: Let us consider the following problems: what is the lowest (highest) average contract length consistent with a particular reset proportion !: Mathematically, we know that since A(!) is non-empty, closed and bounded andT ( )is continuous, both a maximum and a minimum will exist. Turning to the minimization problem …rst: we have to choose 2 F 1 to solve
min T( ) s:t: 2A(!) (14)
Proposition 3 Let 2 F 1 solve (14) to give the shortest average con-tract length T ( ).
(a) No more than two sectors ihave values greater than zero
(b) If there are two sectors i > 0, j > 0 then will be consecutive integers (ji jj= 1).
(b) There is one solution i¤! 1=k 2Z
+. In this case, k = 1. We can sum up the proposition by saying that the shortest average con-tract length consistent with a given proportion of resetters is the simplest
GT E that can represent it. It is either a pure simple Taylor, or an only slightly less simple GT E with two sectors of consecutive lengths. Note that whilst Proposition 3 is derived for aGT E, under the equivalence established by Proposition 1, it will also hold across allGCs. It is a distribution which minimises mean contract length, and this distribution can be seen as either aGT E or aGC.
We can also ask what is themaximum average contract length consistent with a proportion of resetters:
max T ( ) s:t: 2A(!)
Proposition 4 Given the longest contract duration F, the distribution of contracts that maximises the average length of contract subject to a given proportion of …rms resetting price is
F = F F 1(1 !) 1 = F F 1! 1 F 1
with i = 0 for i= 2:::F 1: The maximum average contract length is
Tmax =F(1 !) + 1
There will be one contract length with F = 1if!=F 1, and 1 = 1if
! = 1. For all intermediate values, F: 1 >0. This proposition implies that Tmax ! 1as F ! 1.
Let us return to theBK data set in the light of the preceding two propo-sitions. First, we can ask the what is the shortest average contract length which is consistent with the BK data set. The BK data set gives us for each sectork the proportion of …rms changing price: !k in any month: Fol-lowing Bils and Klenow themselves, it is natural to interpret this as aM C, that within each sector there is a Calvo process. This was the assumption we used to generate Figure 2. In terms of the mean contract length in the
BK M C, this is calculated as TM C = n X k=1 k 2 !k !k = 4:4
Where durations are in Quarters unless otherwise speci…ed. The minimum average contract length within each sector is simply!k;so that the minimum average contract length in theBK data is22:
Tmin = n X k=1 k 1 !k = 2:7
Clearly, for any data set like this (based on the proportion of …rms changing price in a period), the shortest average contract length can be achieved using the Taylor model: using the Multiple Calvo yields an average duration nearly twice as long, with the linear relation:
TM C = 2Tmin 1:
The maximum average contract length is not revealed by the data set. There are some sectors with very low percentages of price changing: coin
22Note that this exceeds the level reported by Bils and Klenow! That is becuase they interpret the proportion as arising from a continuous time process. We are adopting the discrete time approach.
operated laundry has 1.2% of prices changing a month implying aminimum23 mean contract length in that sector of 83 months (nearly 7 years). However, the nature of the BK data set says nothing about the distribution within the sectors, so there is no meaningful upper bound on the average length of contracts.
Lastly, let us consider what might happen if we aggregate over all sectors by taking the mean proportion of …rms changing price,
^ ! = n X k=1 k!k
This gives us an estimate of average durationT^: ^
T = 2 ^
! 1 = 2:47
Now, clearly, since contract length is a convex function of !k, by Jensen’s inequalityT^ TM C. If we use the actual BK data, we have!^ = 0:209per
month, yielding the reported estimate of2:47quarters which is just over one half of the "true" M C value. This shows that aggregating over sectors in this way can be extremely misleading and will considerably underestimate the "true " value even if one believes the Calvo story24. However, the M C might not be the true model. Even with this degree of disaggregation, there may well be intra-sectoral variation. Ultimately, what is needed is the individual price data. Certainly, using data of the proportions changing price in a period is of limited value and might not even get you into the right ballpark. That is certainly what the BK data set tells us.
7
Conclusion
In this paper we have developed a consistent and comprehensive framework for analyzing di¤erent pricing models which generate steady state distribu-tions of duradistribu-tions which can be used to understand dynamic pricing mod-els. In particular, the distribution of completed contract lengths across …rms
(DAF) is a key perspective which is fundamental to understanding and com-paring di¤erent models. Any steady state distribution of durations can be
23Recall, we have thelower bound onF sinceF (!
k) 1:
looked at in terms of completed durations, which suggests it can be modelled as aGT E; it can also be thought of in terms of Hazard rates which suggests the GC approach. Both the GC and the GC are comprehensive: they can represent all possible steady states. Furthermore, they are closed under ag-gregation. Unlike the simple Calvo and Taylor models, they are consistent with heterogeneity in the economy.
Once we have controlled for the distribution of contract durations, we can compare di¤erent pricing models. The concept Forward Lookingness is useful in comparing the way di¤erent pricing rules put weights on the future periods. We …nd that theGC is less myopic than the correspondingGT E, echoing the …nding of Dixon and Kara (2005a) comparing Calvo and the Calvo-GT E. We then illustrate this by using a standard macromodel using the Bils-Klenow data set, which we interpret (following Bils and Klenow) as an M C process. We see even though the distributions are identical, the three pricing models are di¤erent. TheGC andM C are close to each other and had the same forward lookingness. TheGT E is more myopic and has a di¤erent impulse-response in relation to a monetary shock. In particular, for particular parameterization, the GT E can display a hump-shaped in‡ation response, whereas theGC andM C never have a hump.
The analysis also has implications for how we interpret the data on the proportion of …rms setting a price in a particular period. The minimum average length consistent with this is given by the simplest GT E. There is no reasonable upper bound unless we have an upper bound on the maximum contract length possible. Certainly, there are severe problems of aggregation which indicate that using such data might lead very inaccurate estimates of average contract length. The analysis also indicates that existing estimates of price-stickiness are biased downwards and that in reality prices are stickier than some have maintained.
Our analysis has looked at one type of wage or price setting behaviour: the contract consists in the setting of a nominal price or wage that persists throughout the contract. As we show in Dixon and Kara (2006b), other types of price and wage setting can be dealt with in this framework. For example, we can have Fischer contracts, where …rms or unions set a trajectory of nominal prices over the life of the contract. This is essentially the approach taken by Mankiw and Reis (2002). There are other possibilities such as indexation during the contract once the ini
